ABCD is a parallelogram, P and Q are the points on the diagonal AC such that `AP = QC`, then quadrilateral BPDQ is a :

Options

A. trapezium
B. parallelogram
C. square
D. none of these

Answer

To solve the problem, we need to analyze the properties of the parallelogram ABCD and the points P and Q on the diagonal AC.

### Step-by-Step Solution:

1. **Identify the Parallelogram**:
- We have a parallelogram ABCD. In a parallelogram, opposite sides are equal and parallel.
- Let’s denote the vertices as follows: A (top left), B (top right), C (bottom right), D (bottom left).

2. **Draw the Diagonal AC**:
- Draw the diagonal AC. This diagonal divides the parallelogram into two triangles: ΔABC and ΔADC.

3. **Locate Points P and Q**:
- Points P and Q are located on diagonal AC such that the lengths AP and QC are equal.
- This means that if we let AP = x, then QC = x as well.

4. **Determine the Lengths**:
- Since AP = QC, we can denote the length of AC as the total length (let’s say it is L).
- Therefore, we have:
- AC = AP + PQ + QC
- L = x + PQ + x
- L = 2x + PQ
- From this, we can express PQ as:
- PQ = L - 2x.

5. **Connect Points B, P, D, and Q**:
- Now, we will connect points B to P and D to Q to form the quadrilateral BPDQ.
- We also connect B to D to complete the quadrilateral.

6. **Analyze Quadrilateral BPDQ**:
- We need to determine the properties of quadrilateral BPDQ.
- Since AP = QC, the segments BP and DQ will also be equal in length due to the properties of the parallelogram (opposite sides are equal).

7. **Check for Parallelism**:
- Since AB is parallel to CD (properties of parallelogram) and BP is equal to DQ, we can conclude that:
- BP || DQ (because they are both segments connecting points on the same diagonal).
- Therefore, we have two pairs of opposite sides that are equal and parallel.

8. **Conclusion**:
- Since both pairs of opposite sides are equal and parallel, quadrilateral BPDQ is a parallelogram.

### Final Answer:
Quadrilateral BPDQ is a **parallelogram**.

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